Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![**Question:**
Find the exponential function of the form \( f(x) = ab^x \) for which \( f(-3) = 7 \) and \( f(0) = 2 \).
---
This problem requires finding the exponential function with the given conditions. The general form of the function is \( f(x) = ab^x \). Two specific points on the function's curve are provided: \( f(-3) = 7 \) and \( f(0) = 2 \).
To solve this, follow these steps:
1. Recognize that \( f(0) = ab^0 \). Since \( b^0 = 1 \), this simplifies to \( f(0) = a \). Given \( f(0) = 2 \), we find:
\[
a = 2
\]
2. Next, use the point \( f(-3) = 7 \):
\[
f(-3) = ab^{-3} = 7
\]
Substitute \( a = 2 \):
\[
2b^{-3} = 7
\]
Solving for \( b \):
\[
b^{-3} = \frac{7}{2} \implies b^3 = \frac{2}{7}
\]
\[
b = \left( \frac{2}{7} \right)^{1/3}
\]
3. Therefore, the exponential function is:
\[
f(x) = 2 \left( \left( \frac{2}{7} \right)^{1/3} \right)^x
\]
This results in the desired function \( f(x) \) that meets the given conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe2c5c293-7ff8-490b-b548-13e2f43ca7a8%2F5003320b-3624-4c1a-b66a-ddcedb195293%2Flhncbf.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
Find the exponential function of the form \( f(x) = ab^x \) for which \( f(-3) = 7 \) and \( f(0) = 2 \).
---
This problem requires finding the exponential function with the given conditions. The general form of the function is \( f(x) = ab^x \). Two specific points on the function's curve are provided: \( f(-3) = 7 \) and \( f(0) = 2 \).
To solve this, follow these steps:
1. Recognize that \( f(0) = ab^0 \). Since \( b^0 = 1 \), this simplifies to \( f(0) = a \). Given \( f(0) = 2 \), we find:
\[
a = 2
\]
2. Next, use the point \( f(-3) = 7 \):
\[
f(-3) = ab^{-3} = 7
\]
Substitute \( a = 2 \):
\[
2b^{-3} = 7
\]
Solving for \( b \):
\[
b^{-3} = \frac{7}{2} \implies b^3 = \frac{2}{7}
\]
\[
b = \left( \frac{2}{7} \right)^{1/3}
\]
3. Therefore, the exponential function is:
\[
f(x) = 2 \left( \left( \frac{2}{7} \right)^{1/3} \right)^x
\]
This results in the desired function \( f(x) \) that meets the given conditions.
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